Billy Joel’s New York state of mind has taken him downtown to 211 Elizabeth St. in Nolita, where he just bought a $3.3 million apartment. For a superstar like Joel, the two-bedroom, 1,600-square-foot condo is a pretty modest purchase, and it looks like the musician is buying it for his daughter, Alexa Ray Joel. She was spotted moving out of her West Village digs in January.

We hope this means that things are looking up for Alexa Ray, a talented singer-songwriter herself, who has said that she wants to help girls combat “heartbreak-related depression,” something she’s experienced. She was hospitalized after downing a handful of pills in December. Sources close to her said she was troubled after breaking up with musician boyfriend Jimmy Riot. But her life seems to have improved since: She will be featured in Prell shampoo ads like her mom, Christie Brinkley, did almost 25 years ago.

There are still three units on the market, priced between $2.35 million and $2.45 million, at 211 Elizabeth St. The 15-unit new development was created by design firm Roman & Williams.

Oh. My. Tod.

Dashing Diego Della Valle, chairman and CEO of fancy footwear company Tod’s, just made headlines for raising his stake in Saks to 15 million shares — 9.4 percent of the company. But his buying spree didn’t end there. Della Valle has also, on the sly, bought a shiny new condo in Miami.

He paid $1.56 million for the three-bedroom, three-bath unit at the Caribbean, located at 3737 Collins Ave. in Miami Beach, says broker Diane Lieberman, whose company, SBI Realty, is the exclusive sales agent. (Lieberman’s son-in-law is Niche Media’s Jason Binn, a publisher of luxury magazines including Gotham.)

Della Valle — who recently recruited Karl Lagerfeld to resuscitate Tod’s once-hot shoe brand, Hogan — runs a company based in Italy. But when he’s in New York, he bunks at his home at 240 Centre St., the former police building in Little Italy.